Journal of Fractional Calculus and Nonlinear Systems

Mathematical Modeling and Analysis of the Spread of Varicella (Chickenpox) Disease with the Caputo Operator

Mubarak Tijani — 2025-06-25

In this study, we present a mathematical model for studying the dynamics of varicella disease transmission incorporating Caputo's fractional derivative to account for memory effects in disease spread. The aim of this research is to construct a robust mathematical framework that combines fractional calculus with disease modeling and explore fractional order parameters impacts on diseases propagations. Thus, the significant contribution to knowledge is the improved understanding of infectious diseases? dynamism as influenced by memory effects thus providing profound insights which can be used as cornerstones in developing preventive measures. The technique of this study relies on integration a system of differential equations which are subjected to the Caputo fractional operator. A variety of data sources were used, including existing epidemiological varicella literature for model calibration. We begin by analyzing the existences and uniqueness of the solutions to this model, utilizing the Banach Fixed-point theorem, the linear stability analysis of the model, using Lyapunov Functional approach. The analysis also includes key epidemiological parameters ($R_0$: basic reproduction number) and dynamic response of the disease-free equilibrium. The effect of the fractional order parameter is investigated through numerical simulations performed by appropriate computational tools especially concerning an infection peak and asymptotic behavior. The results indicate that the varicella transmission dynamics are related to order parameter of fractional order $\alpha$. The results of this study demonstrated that to better describe epidemiological data, it is essential to introduce fractional order operators within infectious disease models able to represent nonlocal and complex phenomena.

Variable order R-L fractional calculus and its Applications

Sayali Nikam — 2025-06-25

This paper presents a concise study of variable-order fractional calculus using the Riemann-Liouville approach. Specifically, we consider the Mittag-Leffler function with a single parameter as the order for both Riemann-Liouville fractional differentiation (FD) and fractional integration (FI). The study explores the impact of varying the parameter in the Mittag-Leffler (M-L) function and applies this variable-order fractional operator to polynomial functions of different degrees. For clarity and completeness, the behavior of the Mittag-Leffler-based Riemann-Liouville fractional calculus is examined both theoretically and graphically.

Existence Results for Implicit Fractional Differential Equations with Riesz-Caputo Derivative

Wafaa Rahou — 2025-06-25

This article investigates the existence and uniqueness of solutions for a class of initial value problems involving implicit fractional differential equations with the Riesz–Caputo fractional derivative. By employing fixed point theorems in conjunction with the technique of measures of noncompactness, we establish key existence and uniqueness results. Furthermore, we demonstrate that the proposed problem exhibits Ulam stability. To support and illustrate our theoretical findings, several examples are provided.

First Step to Spectral Theory with Generalized M Derivative and Applications

Merve Karaoglan — 2025-06-25

In this article, several fundamental spectral results are established for the Sturm–Liouville problem with discrete boundary conditions involving the generalized M-derivative. The paper is organized into four sections. The first section provides a brief historical background of the topic. The second section presents essential definitions and foundational theorems. In the third section, we investigate the uniqueness theorem for the generalized M-derivative Sturm–Liouville boundary value problem on a finite interval and offer two distinct methods for representing the solution. The final section offers a comprehensive evaluation of the study, including a detailed visual analysis using graphical illustrations.